#352556

order	512
nilpotency class	3
isoclinism	isoclinic class of #347788
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

575 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6)(3,7,8,5)(11,17)(12,19)(14,20)(16,22)(18,23)(21,24)(25,26)(27,30)(28,31)(29,33)(32,36)(34,37)(35,38)(39,40);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(25,27)(26,30)(28,35)(29,34)(31,38)(32,39)(33,37)(36,40);(9,11,12,17,13,18,19,23)(10,14,16,20,15,21,22,24)(27,34)(30,37)(35,40)(38,39);(10,15)(14,21)(16,22)(20,24)(26,31)(28,36)(30,38)(32,33)(35,40)(37,39)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=f,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,c^b=ci,d^b=dh,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259752);